How to compute the max/min surface area of a donut-shape solid generated by a revolved 2D circle, as the volume of the solid doesn’t change?

A donut shape solid is generated by revolving a circle $ (x-a)^2+y^2=b^2$ around the y-axis. $ a$ is the distance from the center of the hole of the donut to the center of the circle revolved, and $ b$ is the radius of the circle revolved. I’m trying to compute the minimized surface area and the maximized surface area (if the solid has) with the value of $ a$ and $ b$ , while the volume of the solid doesn’t change (which is $ 90\pi^2$ ). Thanks a lot if someone can help me 🙂

Is the star chart a material component for the Guidance and Guiding Bolt spells from the Circle of Stars druid’s Star Map feature?

The Circle of Stars druid’s Star Map feature grants the following benefits (TCoE, p. 38):

You’ve created a star chart as part of your heavenly studies. It is a Tiny object and can serve as a spellcasting focus for your druid spells. You determine its form by rolling on the Star Map table or by choosing one.

While holding this map, you have these benefits:

  • You know the guidance cantrip.
  • You have the guiding bolt spell prepared. It counts as a druid spell for you, and it doesn’t count against the number of spells you can have prepared.
  • You can cast guiding bolt without expending a spell slot. You can do so a number of times equal to your proficiency bonus, and you regain all expended uses when you finish a long rest.

Neither guiding bolt nor guidance have a material component. It would seem from the language that RAW one must hold the chart in one hand and cast these spells with the other, unless one has the War Caster feat.

Is my interpretation of the rules correct? And what is actually intended?

Is the star chart a material component for guidance and guiding bolt?

Can a Circle of the Stars Druid roll a natural d3 (or other odd-sided die) to bias their Cosmic Omen roll?

The sixth level Circle of the Stars feature, Cosmic Omen, allows the druid to, for the remainder of a day, use their reaction to either help their allies or hinder their foes:

Whenever you finish a long rest, you can consult your Star Map for omens. When you do so, roll a die. Until you finish your next long rest, you gain access to a special reaction based on whether you rolled an even or an odd number on the die:

Weal (even). Whenever a creature you can see within 30 feet of you is about to make an attack roll, a saving throw, or an ability check, you can use your reaction to roll a d6 and add the number rolled to the total.

Woe (odd). Whenever a creature you can see within 30 feet of you is about to make an attack roll, a saving throw, or an ability check, you can use your reaction to roll a d6 and subtract the number rolled from the total.

Cosmic Omen, Tasha’s Cauldron of Everything, pg. 38

Note, however, that in the description of this feature, the player is simply instructed to roll "a die", without qualifying what kind of die it has to be. This poses no issue for the "standard issue" dice used in 5e D&D, since the d4, d6, d8, d10, d12, and d20 all have an even number of sides, and therefore an [assumed, with fair dice] equal probability of rolling an even or odd number.

However, there are dice that have odd numbers of sides. In this situation, a player could pick up a three-sided die, for example, and they’d have a 2/3rds chance of rolling an odd number, biasing their feature towards hindering enemies.

Is there any rule in 5th Edition D&D that forbids a player from choosing to do this?

Tasha’s Druid Circle of the Stars Cosmic Omen

I have a question about Circle of the Stars 6th level feature:

Cosmic Omen
When you reach 6th level, you learn to use your star map to divine the will of the cosmos. Whenever you finish a long rest, you can consult your Star Map for omens. When you do so, roll a die. Until you finish your next long rest, you gain access to a special reaction based on whether you rolled an even or an odd number on the die:
Weal (even). Whenever a creature you can see within 30 feet of you is about to make an attack roll, a saving throw, or an ability check, you can use your reaction to roll a d6 and add the number rolled to the total.
Woe (odd). Whenever a creature you can see within 30 feet of you is about to make an attack roll, a saving throw, or an ability check, you can use your reaction to roll a d6 and subtract the number rolled from the total.
You can use this reaction a number of times equal to your proficiency bonus, and you regain all expended uses when you finish a long rest.

I’m not sure whether I have to use my reaction before I know the score the enemy/party member rolled or I can do it after I know what their score is.

Demon of the First Circle limitations

Demon of the First Circle or any of the Demon spells seems ripe for exploitation. I’m wondering how other storytellers have dealt with this issue.

I’m worried because of the ease of summoning, and the length the demon will serve you. The way I read the rules there is nothing preventing a sorcerer from summoning one demon every night. The risk of failing Int + Occult vs the demons resolve is also very small. Spend willpower and a stunt, and you’re almost guaranteed to succeed. That is before taking into consideration that most sorcerers will have high int and occult, and possibly an excellency to boost the roll if they feel like it. Once summoned the demon is your slave for a year and a day.

Given the system, it feels like most sorcerers should be surrounded by a small army of demons to do their bidding.

How have other storytellers dealt with this issue? Is there anything from e3 or the other editions that can be used as a guide to reduce the exploitability of the summon spells?